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3 amplitude and phase

4 coherence and pixel matching

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6 resolution: optical vs. microwave D s = 2H sinθ r = 2H λ L H = 800km. Optical : L = 1m λ = 0.5µm D s = 0.8m Microwave : L = 10m λ = 0.23m D s = 46,000m!!!!!!

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8 2-D Aperture Flight Path Ground Track Azimuth Range ' igure'5:'radar'range'and'azimuth'dimension L /2 W /2 ( ) = A(x, y) P θ x,θ y exp i 2π ( λ xsinθ + ysinθ x y L /2 W /2 & ' ( ) ) * + dxdy P θ x,θ y $ ( ) = LW sinc πw sinθ x & % λ ' $ ( ) sinc π L sinθ y % & λ ' ( )

9 range resolution θ H ρ τ θ - look angle H - spacecraft height τ - pulse length C - speed of light (sound) R r = Cτ 2sinθ

10 azimuth resolution L θ r ρ R a L - length of radar antenna ρ - nominal slant range H/cosθ λ - wavelength of radar unfocussed R a = ρ sinθ r = ρλ/ L R a ' = focussed λh 2R a cosθ = L 2

11 PRF - upper bound The PRF cannot be too large or the return pulses from the near range and far range will overlap in time as shown in Figure A9. H θ 1 θ 2 near far range range Figure A9. End view of the distance to the near range and far range of the radar illumination pattern. 1 PRF < t 2 t 1 t 1 = 2H C cosθ 1 t 2 = Δt = 2H ( C secθ secθ 2 1) 2H C cosθ 2 (A12) 2V L < PRF < C ( 2H secθ secθ 2 1) 1 For ERS the look angles to the rear range and far range are 18 and 24, respectively. Thus the maximum PRF is 4777 Hz. The actual PRF of 1680 is safely below this value.

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15 How is the image focused in the computer?

16 SAR processor

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19 SAR processor

20 Range Compression

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22 Azimuth compression Azimuth compression or azimuth focusing involves coherent summation of echos at a constant range from the point reflector. The geo metry of the strip-mode acquisition is shown in Figure B6 s H R o R(s) x V(s-s o ) Figure B6. Geometry of radar passing over a point reflector where V the effective speed which is about equal to the ground track speed s slow time along the satellite track s o time when the center of the radar echo passes over the point reflector R o = R near + n * (C / fs) minimum range from the spacecraft to the target near range to first data sample in the swath R near

23 The range to the point reflector evolves with time as R 2 ( ) 2. ( s) = R 2 o + V 2 s s o The complex phase of the return echo is $ C(s) = exp i 4π % & λ R(s) ' ( ). The range versus slow time is approximately a hyperbola but for mathematical convenience we'll approximate this using a parabola R(s) = R o + R R o (s s o ) + o 2 (s s o) where the dot indicates derivative with respect to slow time, s. Curlander and McDonough [1991] discuss the accuracy of this polynomial approximation and it is also discussed below in terms of the ALOS SAR. The approximation is good enough for strip-mode SAR but may be inadequate for the much longer apertures associated with spotlight-mode SAR. Now we can write the phase of t he return signal as a function range, range rate, and range acceleration. + C(s) = exp, i 4π $ λ % R + o R o s s o - ( ) + R o ( s s o ) 2 / 2 '. (/ 0 It is more common to describe the parameters for focusing the SAR image as the Doppler centroid f Dc and the Doppler frequency rate f R. The relationships are: f Dc = 2 R λ and f R = 2 R λ.

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25 Length of synthetic aperture - The length of the synthetic aperture L a depends on the length of the radar ground footprint in the azimuth direction, which is approximately L a = R o λ L where L is the physical length of the antenna, λ is the wavelength and R o is the slant range given above. The length of the aperture in terms of radar echos is given by (B n a = L a PRF V (B This is simply the length of the synthetic aperture divided by the along-track sampling distance. The following table provides these quantities for SARs of interest for interferometry today. Table B3. Length of synthetic aperture for three satellites. V λ PRF R o L L a n a n a (m/s) (m) (Hz) (km) (m) (m) theory actual ERS-1/ ~ Envisat ~ ALOS ~ ,

26 phase history of point reflector ' C(s) = exp ( i 4π ) λ # $ R( s) * %& +, parabolic approximation to range history R(s) = R o + R o (s s o ) + R o 2 (s s o) f DC = 2 R λ f R = 2 R λ Least-squares fit of range history for each point in DEM provides both the [ ] accurate position in range azimuth R o,s o space and the Dopper centroid and rate parameters needed to focus the image. This analysis only needs to be applied to the master image.

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29 SAR processor

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31 amplitude image

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